Sufficient condition for rectifiability involving Wasserstein distance $W_2$

Abstract

A Radon measure $\mu$ is $n$-rectifiable if it is absolutely continuous with respect to ${\mathcal{H}}^n$ and $\mu$-almost all of supp$\mu$ can be covered by Lipschitz images of ${\mathbb{R}}^n$. In this paper we give two sufficient conditions for rectifiability, both in terms of square functions of flatness-quantifying coefficients. The first condition involves the so-called $\alpha$ and ${\beta }_2$ numbers. The second one involves ${\alpha }_2$ numbers – coefficients quantifying flatness via Wasserstein distance $W_2$. Both conditions are necessary for rectifiability, too – the first one was shown to be necessary by Tolsa, while the necessity of the ${\alpha }_2$ condition is established in our recent paper. Thus, we get two new characterizations of rectifiability.